#8934828

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#59303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,4)(3,9,5,8)(6,12,14,10)(11,15,13,16)(17,18,23,26)(19,24,27,32)(20,30,29,22)(21,31,28,25)(33,34)(35,40)(36,37)(38,46)(39,41)(42,43)(44,47)(45,48);(1,3,7,5)(2,8,4,9)(6,11,14,13)(10,15,12,16)(17,19)(18,24)(20,21)(22,25)(23,27)(26,32)(28,29)(30,31)(33,35)(34,40)(36,44)(37,47)(38,42)(39,45)(41,48)(43,46);(2,4)(6,14)(8,9)(11,13)(19,27)(20,29)(22,30)(24,32)(34,41)(35,45)(36,43)(38,47);(1,4,7,2)(3,8,5,9)(6,12,14,10)(11,15,13,16)(17,20)(18,22)(19,28)(21,27)(23,29)(24,31)(25,32)(26,30)(33,36)(34,42)(35,46)(37,41)(38,48)(39,43)(40,47)(44,45);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,21,23,28)(18,25,26,31)(19,29,27,20)(22,24,30,32)(33,37,39,42)(34,43,41,36)(35,47,45,38)(40,46,48,44);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,22)(18,20)(19,25)(21,24)(23,30)(26,29)(27,31)(28,32)(33,38,39,47)(34,44,41,46)(35,42,45,37)(36,48,43,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842024);